Mrs. Gomes Found That 40 % 40\% 40% Of Students At Her High School Take Chemistry. What Is The Probability That Exactly 4 Students Have Taken Chemistry? Round The Answer To The Nearest Thousandth.A. 0.005 B. 0.008 C. 0.213 D. 0.227

Introduction

In this article, we will explore the concept of probability and how it can be applied to real-world scenarios. Mrs. Gomes, a high school teacher, has observed that $40\%$ of her students have taken chemistry. We will use this information to calculate the probability that exactly 4 students have taken chemistry.

Understanding Probability

Probability is a measure of the likelihood of an event occurring. It is usually expressed as a value between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. In this case, we want to find the probability that exactly 4 students have taken chemistry.

The Binomial Distribution

The binomial distribution is a probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success. In this case, we can use the binomial distribution to model the number of students who have taken chemistry.

Calculating the Probability

Let's assume that we have a total of $n$ students in the high school. We know that $40\%$ of the students have taken chemistry, so the probability of a student taking chemistry is $p = 0.4$. We want to find the probability that exactly 4 students have taken chemistry.

We can use the binomial distribution formula to calculate this probability:

$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

where $X$ is the number of students who have taken chemistry, $k$ is the number of students who have taken chemistry (in this case, 4), $n$ is the total number of students, and $p$ is the probability of a student taking chemistry.

Solving for n

We are given that $40\%$ of the students have taken chemistry, so we can set up the equation:

$$\frac{40}{100} = \frac{k}{n}$$

We can solve for $n$:

$$n = \frac{100k}{40}$$

$$n = \frac{5k}{2}$$

Substituting Values

We want to find the probability that exactly 4 students have taken chemistry, so we can substitute $k = 4$ into the equation:

$$n = \frac{5(4)}{2}$$

$$n = 10$$

Calculating the Probability

Now that we have the value of $n$, we can calculate the probability that exactly 4 students have taken chemistry:

$$P(X = 4) = \binom{10}{4} (0.4)^4 (0.6)^6$$

$$P(X = 4) = 210 (0.0256) (0.046656)$$

$$P(X = 4) = 0.213$$

Conclusion

In this article, we used the binomial distribution to calculate the probability that exactly 4 students have taken chemistry. We found that the probability is approximately $0.213$. This means that if we were to randomly select 10 students from the high school, the probability that exactly 4 of them have taken chemistry is approximately $0.213$.

Answer

The correct answer is:

C. 0.213

Discussion

This problem is a classic example of how probability can be used to model real-world scenarios. The binomial distribution is a powerful tool for calculating probabilities in situations where there are a fixed number of independent trials, each with a constant probability of success. In this case, we used the binomial distribution to calculate the probability that exactly 4 students have taken chemistry.

Real-World Applications

This problem has many real-world applications, such as:

• Quality Control: In manufacturing, quality control engineers use probability to calculate the likelihood of defects in a product.
• Finance: In finance, probability is used to calculate the likelihood of investment returns and to manage risk.
• Medicine: In medicine, probability is used to calculate the likelihood of disease outcomes and to make informed decisions about treatment.

Conclusion

In conclusion, this problem demonstrates the power of probability in modeling real-world scenarios. The binomial distribution is a useful tool for calculating probabilities in situations where there are a fixed number of independent trials, each with a constant probability of success. We hope that this article has provided a clear understanding of how probability can be used to solve real-world problems.

Introduction

In our previous article, we explored the concept of probability and how it can be applied to real-world scenarios. We used the binomial distribution to calculate the probability that exactly 4 students have taken chemistry. In this article, we will answer some frequently asked questions about the problem.

Q: What is the probability that exactly 4 students have taken chemistry?

A: The probability that exactly 4 students have taken chemistry is approximately 0.213.

Q: What is the binomial distribution?

A: The binomial distribution is a probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success.

Q: How do I calculate the probability that exactly 4 students have taken chemistry?

A: To calculate the probability that exactly 4 students have taken chemistry, you can use the binomial distribution formula:

$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

where $X$ is the number of students who have taken chemistry, $k$ is the number of students who have taken chemistry (in this case, 4), $n$ is the total number of students, and $p$ is the probability of a student taking chemistry.

Q: What is the value of n in this problem?

A: The value of $n$ is 10, which is the total number of students in the high school.

Q: What is the value of p in this problem?

A: The value of $p$ is 0.4, which is the probability of a student taking chemistry.

Q: Can I use the binomial distribution to calculate the probability that more than 4 students have taken chemistry?

A: Yes, you can use the binomial distribution to calculate the probability that more than 4 students have taken chemistry. However, you will need to calculate the probability that 5 or more students have taken chemistry, which can be done using the binomial distribution formula.

Q: Can I use the binomial distribution to calculate the probability that fewer than 4 students have taken chemistry?

A: Yes, you can use the binomial distribution to calculate the probability that fewer than 4 students have taken chemistry. However, you will need to calculate the probability that 0, 1, 2, or 3 students have taken chemistry, which can be done using the binomial distribution formula.

Q: What are some real-world applications of the binomial distribution?

A: The binomial distribution has many real-world applications, such as:

• Quality Control: In manufacturing, quality control engineers use probability to calculate the likelihood of defects in a product.
• Finance: In finance, probability is used to calculate the likelihood of investment returns and to manage risk.
• Medicine: In medicine, probability is used to calculate the likelihood of disease outcomes and to make informed decisions about treatment.

Conclusion

In conclusion, this article has provided answers to some frequently asked questions about the problem of calculating the probability that exactly 4 students have taken chemistry. We hope that this article has provided a clear understanding of how probability can be used to solve real-world problems.

Frequently Asked Questions

• Q: What is the probability that exactly 4 students have taken chemistry? A: The probability that exactly 4 students have taken chemistry is approximately 0.213.
• Q: What is the binomial distribution? A: The binomial distribution is a probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success.
• Q: How do I calculate the probability that exactly 4 students have taken chemistry? A: To calculate the probability that exactly 4 students have taken chemistry, you can use the binomial distribution formula:

$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

• Q: What is the value of n in this problem? A: The value of $n$ is 10, which is the total number of students in the high school.
• Q: What is the value of p in this problem? A: The value of $p$ is 0.4, which is the probability of a student taking chemistry.

Additional Resources

• Binomial Distribution Formula: The binomial distribution formula is:

$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$$

• Binomial Distribution Table: The binomial distribution table is a table that shows the probability of exactly $k$ successes in $n$ trials, where $p$ is the probability of success.
• Probability Calculator: A probability calculator is a tool that can be used to calculate the probability of an event occurring.